Fourier-Stieltjes algebra

Let be an arbitrary locally compact group. For , let , where is the set of all equivalence classes of unitary continuous representations of (cf. also Unitary representation). The completion of with respect to this norm is a Banach algebra, denoted by and called the full -algebra of . If is Abelian and its dual group, then is isometrically isomorphic to the Banach algebra of all complex-valued continuous functions on vanishing at infinity.

Let be the complex linear span of the set of all continuous positive-definite functions on .

1) The -vector space is isomorphic to the dual space of . With the dual norm and the pointwise product on , is a commutative Banach algebra [a4].

This Banach algebra is called the Fourier–Stieltjes algebra of . If is Abelian, then is isometrically isomorphic to the Banach algebra of all bounded Radon measures on .

2) On the boundary of the unit ball of (i.e. on ) the weak topology coincides with the compact-open topology on ([a3]; see also [a9], [a6]).

3) The following properties are satisfied ([a4]):

a) The Fourier algebra is a closed ideal of ;

b) ;

c) coincides with the closure in of ;

d) , with equality of the corresponding norms. Here, is the algebra of functions of compact support on . In [a14], M.E. Walter showed that (and also ) completely characterizes . More precisely, assume that and are locally compact groups; then the following assertions are equivalent:

the locally compact groups and are topologically isomorphic;

the Banach algebras and are isometrically isomorphic;

the Banach algebras and are isometrically isomorphic.

He also gave a description of the dual of .

For a connected semi-simple Lie group , M. Cowling [a1] has given a description of the spectrum of ; surprisingly, if is Abelian, then the spectrum of seems to be much more complicated than in the non-Abelian case!

If is amenable, then ([a3])

(a1)

V. Losert [a12] proved the converse assertion: if (a1) holds, then must be amenable!

In a difficult paper [a7], C.S. Herz tried to extend the preceding results, replacing unitary representations by representations in Banach spaces. He partially succeeded in the amenable case. See also [a2], [a5].

M. Lefranc generalized Paul Cohen's idempotent theorem to for arbitrary locally compact groups ([a10], [a8]; see also [a11] for detailed proofs).

See also Figà-Talamanca algebra.

References

[a1]	M. Cowling, "The Fourier–Stieltjes algebra of a semisimple Lie group" Colloq. Math. , 41 (1979) pp. 89–94
[a2]	M. Cowling, G. Fendler, "On representations in Banach spaces" Math. Ann. , 266 (1984) pp. 307–315
[a3]	A. Derighetti, "Some results on the Fourier–Stieltjes algebra of a locally compact group" Comment. Math. Helv. , 45 (1970) pp. 219–228
[a4]	P. Eymard, "L'algèbre de Fourier d'un groupe localement compact" Bull. Soc. Math. France , 92 (1964) pp. 181–236
[a5]	G. Fendler, "An -version of a theorem of D.A. Raikov" Ann. Inst. Fourier (Grenoble) , 35 : 1 (1985) pp. 125–135
[a6]	E.E. Granirer, M. Leinert, "On some topologies which coincide on the unit sphere of the Fourier–Stieltjes algebra and of the measure algebra " Rocky Mount. J. Math. , 11 (1981) pp. 459–472
[a7]	C. Herz, "Une généralisation de la notion de transformée de Fourier–Stieltjes" Ann. Inst. Fourier (Grenoble) , 24 : 3 (1974) pp. 145–157
[a8]	B. Host, "Le théorème des idempotents dans " Bull. Soc. Math. France , 114 (1986) pp. 215–223
[a9]	K. McKennon, "Multipliers, positive functionals, positive-definite functions, and Fourier–Stieltjes transforms" Memoirs Amer. Math. Soc. , 111 (1971)
[a10]	M. Lefranc, "Sur certaines algèbres sur un groupe" C.R. Acad. Sci. Paris Sér. A , 274 (1972) pp. 1882–1883
[a11]	M. Lefranc, "Sur certaines algèbres sur un groupe" Thèse de Doctorat d'État, Univ. Sci. et Techn. du Languedoc (1972)
[a12]	V. Losert, "Properties of the Fourier algebra that are equivalent to amenability" Proc. Amer. Math. Soc. , 92 (1984) pp. 347–354
[a13]	J.-P. Pier, "Amenable locally compact groups" , Wiley (1984)
[a14]	M.E. Walter, "-algebras and nonabelian harmonic analysis" J. Funct. Anal. , 11 (1972) pp. 17–38